Davies is one of the brightest talents coming through at Goodison Park right now, having been handed his first-team debut during the latter stages of Roberto Martinez's tenure and becoming a more prominent member of the squad while Ronald Koeman was in charge.
He has made 20 Premier League appearances already this season, but has started just 11 times and has not been in the starting lineup too often following Sam Allardyce's appointment as manager.
The Mirror reports that Big Sam is ready to sanction the departures of a number of players within his squad this month and West Ham are keen to see if Davies is one of the players he would be willing to lose.
The Mirror suggests that David Moyes - who was first-team manager of the Toffees when the teenager was signed by the Merseyside club - is keen on adding Davies to his midfield.
It would be no surprise if Davies is not one of the players that Allardyce is looking to move on as he clearly boasts an enormous amount of potential and a move away from Goodison Park would not be well received by the supporters.
However, perhaps the Hammers will be hoping that if Everton struggle to attract bids for some of those in the shop window, Allardyce and the board may be more willing to consider a bid for Davies given that he has not quite been a key man for the Toffees this season.